How do I backup my iPhone with iCloud?

Apple's iCloud servers will automatically back up your iOS device, assuming that iCloud Backup is enabled. To enable iCloud navigate to Settings -> iCloud -> Backup & Storage.

The backup runs automatically every day when your device is:

Connected to the Internet via Wi-Fi
Connected to your charger
Your device screen is locked

You can back up with iCloud manually if you are connected to a Wi-Fi network by selecting Settings -> iCloud -> Storage & Backup -> Back Up Now.

iTunes has a nasty habit of deleting the previous backup every time it does a backup. If you really have an old backup intact, there is a way to restore from it. I forget the procedure, but I'm sure you can Google it. It's a good idea to manually go into your backup directory and manually copy it over to a new name once in a while. That way iTunes won't touch it, and you can use it in an emergency such as yours.

Here's a website that describes how to do it:

iCloud backup is not that safe than we think and the free backup space is limited. Fortunately, we have other choices to backup iPhone:

Here's also an article about backing up locked iPhone.

go to set up as new phone Settings>General>Reset>Erase All Content & Settings general iPhone software update update take about 10min then restart and restore from latest back up with the latest ios7.1.1

Have you tryied to take a look at the iCloud settings to see if you have contacts enabled? Also you can check out the mail settings to see if u have enabled it there too. If nothing don't work, the only way is to restore your phone and load back up your content via iCloud.